MT1002

Method For Finding the Inverse of a Matrix by Column Operations

*You cannot mix row and column operations!

sinh(α)cosh(β) = ?

(1/2)[sinh(α+b)+sinh(α-β)]

1 + 2sinh²x = ?

1 + 2sinh²x = cosh(2x)

1 - tanh²x = ?

1 - tanh²x = sech²x

1/(det(A)) = ?

1/(det(A)) = det(A⁻¹)

2cosh²x - 1 = ?

2cosh²x - 1 = cosh(2x)

2sinh(x)cosh(x) = ?

2sinh(x)cosh(x) = sinh(2x)

Derivative of coth(x)

= -cosech²(x)

Derivative of cosech(x)

= -coth(x)cosech(x)

Derivative of sech(x)

= -tanh(x)sech(x)

Derivative of tanh(x)

= sech²x

Counterexamples

A counterexample to the implication 'if H then C'. This should be a single example (usually a specified mathematical object) that satisfies the hypothesis H but does not satisfy the conclusion C. Usually, a counterexample to 'if H then C' is enough to prove that this implication is not necessarily true. A proof that 'if H then not C' is enough to prove that the implication 'if H then C' is never true. There is a distinct difference between the two cases

How to Prove a Matrix is Invertible Using Determinants

A is invertible if det ≠ 0.

Proofs

A proof of an implication is a series of logical steps that start with the hypothesis H of the result and end with the conclusion C.

Derivative of sinh⁻¹(x)

All real numbers

L'Hôpital's Rule

Can also be applied to ∞/∞

Inverse Matrices

Let A be an n × n square matrix. If there exists an n × n square matrix B such that AB = Iₙ = BA, then A is called invertible, and B is called an inverse of A. The inverse of A is denoted by A⁻¹. (1) For A−1 to exist, A must be square. (2) If A is square, A−1 sometimes exists, but not always

Complex Number Real Values Notation

Re(z)

Principal Argument of Complex Numbers

The unique argument θ ∈ arg(z) with -π < θ ≤ π...is denoted by Arg(z).

Hyberbolic Cosecant (cosech(x))

We can define the hyperbolic cosecant, cosech (csch), to be 1/sinh(x) = 2/[(e^x) - (e^-x)]

Hyberbolic Secant (sech(x))

We can define the hyperbolic secant, sech, to be 1/cosh(x) = 2/[(e^x) + (e^-x)]

Hyperbolic Cotangent (coth(x))

We can define the hyperbolic tangent, coth, to be 1/tanh(x) = [[(e^x) + (e^-x)]/[(e^x) - (e^-x)]

Multiplication of Two Complex Numbers in Polar Form

arg(zw) = arg(z) + arg(w)

cosh(2x) = ?

cosh(2x) = cosh²x+sinh²x = 2cosh²x - 1 = 1 + 2sin²x

cosh(x±y) = ?

cosh(x±y) = cosh(x)cosh(y)±sinh(x)sinh(y)

cosh²(x) = ?

cosh²(x) = (1/2)(1 + cosh(2x)) = 1 + sinh²(x)

cosh²x + sinh²x = ?

cosh²x + sinh²x = cosh(2x)

cosh²x - sinh²x = ?

cosh²x - sinh²x = 1

det Iₙ = ?

det Iₙ = 1

What is the Determinant of a Triangular Matrix?

det(A) = a₁₁a₂₂a₃₃...aₙₙ

det(AB) = ?

det(AB) = det(A) det(B)

Even Functions

f(-x) = f(x) → Reflected over the y-axis *Functions can be even, odd or neither (in fact most functions are neither), while the only function that is both even and odd is the function f(x) = 0.

sinh(2x) = ?

sinh(2x) = 2sinh(x)cosh(x)

sinh(x±y) = ?

sinh(x±y) = sinh(x)cosh(y) ± cosh(x)sinh(y)

sinh² = ?

sinh²(x) = (1/2)(-1 + cosh(2x)) = cosh²(x) - 1

Derivative of cosh⁻¹(x)

x > 1

Polar Form of a Complex Number

z= |z|(cosθ + isinθ)

de Moivre's Formula

zⁿ = (cosθ + isinθ)ⁿ = cos(nθ) + isin(nθ)

Derivative of tanh⁻¹(x)

−1 < x < 1

Gaussian Elimination

(1) Construct the augmented matrix associated with the system (2) Perform row operations on the augmented matrix until the n×n left hand side is in upper triangular form (i.e. all entries below the main diagonal are 0). If possible try to get 1's in the main diagonal (warning: this is not always possible) (3) Once in upper triangular form, translate the new augmented matrix back into the equations it represents and solve, working from the bottom up,

The Row and Column Operation Theorem

(1) If B is the matrix that results when a single row (column) of A is multiplied by a scaler λ, then det(B) = λ det(A), for all i. (2) If B is the matrix that results when two rows (columns) of A are interchanged, then det(B) = − det(A), for all i, j with i 6= j. (3) If B is the matrix that results when a multiple of one row of a is added to another row, then det(B) = det(A), for all i, j.

Three Elementary Column Operations

(1) Multiplying a column by a non-zero number λ (2) Add a multiple of one column to another column (3) Interchanging two columns

Method For Finding the Inverse of a Matrix by Row Operations

*You cannot mix row and column operations!

Scalar

A scalar is a quantity which has magnitude only. Examples of scalar quantities are mass, area, volume and distance.

Surjective Functions

A surjective function is "onto," which means that every output has at least one possible input (the range of the function is all real numbers)

Vector

A vector is a quantity which possesses both magnitude and direction. Examples of vector quantities are force, velocity and acceleration. These have both magnitude and direction associated with them. Note: Vectors need not be limited to three dimensions. However, in this course we restrict ourselves to vectors in at most three dimentions.

How to Find the Equation of a Plane Given 3 Points

AB x BC, and then plug in (x-x₁), (y-y₁), (z-z₁) for i, j, and k.

Row Matrix (row vector)

An 1 × n matrix

Injective Functions

An injective function is one-to-one, which means that it passes the horizontal line test (can become an inverse function without restricting the domain)

Column Matrix (column vector)

An m x 1 matrix

Bijective Functions

Bijective functions are functions that are both injective (one-to-one) and surjective (onto) over the relevant domain.

Hyberbolic Tangent (tanh(x))

By analogy with the circular functions we may define the hyperbolic tangent tanh (pronounced 'tansh') as sinh(x)/cosh(x) = [[(e^x) - (e^-x)]/[(e^x) + (e^-x)] *This is an odd one-to-one function with horizontal asymptotes at y = 1 and y = −1.

Hyberbolic Functions' Relation to Circular Functions

Care is needed as there is often a sign difference between circular and hyperbolic function identities. Osborn's Rule encapsulates this. It states that a trigonometric identity can be converted to a hyperbolic identity by changing the sign wherever the product of two hyperbolic sine terms occurs

Determinants

Determinants are numbers got by doing certain calculations with the entries of a matrix. They have nice properties and are sometimes helpful when dealing with inverses of a matrix. As with inverses, they are only defined for square matrices.

Disproofs

Disproofs can come in two forms. - Counterexamples - A proof of the implication if H then not C

Odd Functions

Functions that are symmetric to (reflected over) the origin → f(-x) = -f(x) *Functions can be even, odd or neither (in fact most functions are neither), while the only function that is both even and odd is the function f(x) = 0.

Matrices Addition

If A and B are matrices of the same size, then the sum A + B is the matrix obtained by adding the entries of A to the corresponding entries of B.

Scalar Multiplication of Matrices

If a matrix is multiplied by a scalar λ, then every entry in the matrix is also multiplied by λ.

Complex Number Imaginary Values Notation

Im(z)

Direct Proof

In a direct proof, you can start with the hypothesis H and aim straight for the conclusion C. Usually, you will use definitions and previously proved results in a direct proof. The final statement may be called a theorem.

Matrix Multiplication

Let A (m x p) and B (p x n) be matrices such that the number of columns in A = the number of rows in B... the product AB of A and B is now defined as an m x n matrix. - AB will not usually be equal to BA - You cannot always cancel in matrix equations. In other words, AB = AC and A ≠ O do not imply that B = C. - A(BC) = (AB)C - A(B + C) = AB + AC - (λA)B = λ(AB) = A(λB) - OA = AO = O - If A is n × n and I is the n × n identity matrix, then AI = IA = A.

Matrix

Let n, m be positive integers. An m × n matrix A is a rectangular array of numbers with m rows and n columns.

Implication

Many ways to write an implication: - H implies C - H → C - C is implied by H - If H then C - C if H - C holds if H holds

Derivative of cosh(x)

Note: Osborn's rule does not apply to the calculus results.

Derivative of sinh(x)

Note: Osborn's rule does not apply to the calculus results.

Osborn's Rule

Osborn's rule is a rule for converting a trigonometric identity into a corresponding hyperbolic one. The rule states that one replaces every occurrence of sine or cosine with the corresponding hyperbolic sine or cosine, and wherever one has a product of two sines, the product of the hyperbolic sines must be negated.

Three Elementary Row Operations

Row operations are very helpful in calculating the inverse A⁻¹ of a given matrix A provided the inverse exists. (1) Multiplying a row by a non-zero number λ (2) Add a multiple of one row to another row (3) Interchanging two rows

Converse (Reverse Direction)

The converse of an implication may or may not be true.

Cramer's Rule

The determinant of A can be computed by expansion along the jth column or the ith row.

Limits at Infinity

The horizontal asymptote of the equation going left or right depending on ±∞.

Hyperbolic Cosh (cosh(x))

The hyperbolic cosine cosh is defined as cosh(x) = [(e^x) + (e^-x)]/2. *Observe that cosh(x) ≥ 1 for all x.

Hyberbolic Sine (sinh(x))

The hyperbolic sine sinh (pronounced 'shine' or sometimes 'sinch' to rhyme with grinch) is given by sinh(x) = [(e^x) - (e^-x)]/2. *We see that sinh(x) is an odd one-to-one function

What Hyperbolic Substitution Can Be Used in √(u² - a²)?

Try substituting u = acosh(θ) (and use the identity cosh²(θ) - sinh²(θ) = 1).

What Trigonometric Substitution Can Be Used in √(a² - u²)?

Try substituting u = asin(θ) or u = acos(θ) (and use the identity cos²(θ) + sin²(θ) = 1).

What Hyperbolic Substitution Can Be Used in √(u² + a²)?

Try substituting u = asinh(θ) (and use the identity cosh²(θ) - sinh²(θ) = 1).

What Trigonometric Substitution Can Be Used in a² + x²?

Try substituting u = atan(θ) (and use the identity 1+tan²θ = sec²θ)

What Hyperbolic Substitution Can Be Used in (u² - a²)?

Try substituting u = atanh(θ) (and use the identity 1−tanh²θ = sech²θ).

e^(-iθ) = ?

e^(-iθ) = cosθ - isinθ

Series Expansion of e^(iθ)

e^(iθ) = cosθ + isinθ